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The Distributive Property is a fundamental principle in algebra that simplifies expressions by distributing multiplication over addition or subtraction. When you have an expression such as \( a(b + c) \), it means you multiply \( a \) by every term inside the parentheses. For example, \( a(b + c) = ab + ac \).
Applying this property is like spreading the multiplication step-by-step over each term inside the brackets.

• Start by multiplying the term outside the brackets with the first term inside.
• Then, do the same with every other term inside the parentheses.

In our exercise, we see this in action: \( 5(3x - 2) \) becomes \( 5 \times 3x - 5 \times 2 \). By doing this, we ensure that the 5 multiplies both terms inside the parentheses, resulting in the expression \( 15x - 10 \). When solving algebraic expressions, mastering the distributive property makes the process much smoother.

Simplifying expressions is the process of making a mathematical expression as straightforward as possible. This doesn't change the value of the expression but makes it easier to work with and understand. To simplify, follow these steps:

• First, ensure that repeated operations, like the use of the distributive property, are fully utilized.
• Remove any parentheses by performing the operations on them.

After you apply the distributive property, like in our exercise \( 5(3x - 2) + 12x \), you clear the brackets and come up with \( 15x - 10 + 12x \).
Next comes the crucial step of combining like terms, simplifying further to make the expression as clear and concise as possible. The goal is to whittle down the expression to its simplest form, making it easier for future calculations.

Combining like terms is an essential step in simplifying algebraic expressions. Like terms are terms that have identical variable parts—it doesn't matter what their coefficients are. For instance, \( x \) terms can only be combined with other \( x \) terms.
This concept helps to reduce expressions into simpler forms. In the context of the exercise, you initially have the expression \( 15x - 10 + 12x \). Identifying the like terms, \( 15x \) and \( 12x \), allows you to combine them, essentially adding their coefficients together.

Sum the coefficients of like terms.

This step removes redundancy, making the expression more manageable.

In this case, \( 15x + 12x \) equals \( 27x \), simplifying the expression to \( 27x - 10 \).
Combining like terms efficiently simplifies expressions and prepares them for further evaluation or computation. Remember, only terms with the same variable and power can be combined. This step is key to mastering algebraic manipulations!